CHAPTER XXX
COMPLEX NUMBERS

242. TAKE two rectangular axes OX and OY, and let the unit i be repre-
sented by a vector of unit length in direction OX. Then any other ordinary
integral or fractional number, such as 2, 3-5, xis represented by a multiple
of the unit vector^ that is, a vector of length 2, 3-5, x in direction OX.

FIG. 201.

FiG. 302.

Let /denote a vector of unit length in direction OY. Then any number
of the form 2/, 3-5*", yi is represented by a vector of length 2, 3-5, y in
direction OY.

Any other vector OP may be expressed as the sum of two vectors in
directions OX and OY.

Thus, in Fig. 201, OP = /• = x -f /V, the 4- sign here denoting vector
addition.

EXAMPLE.— In Fig. 202,

OP = 3 •+ 4}

If the vector r makes an angle 0 with OX3 then

x = r cos 0, y = r sin 0

and 7-0 = x -r- zj = r(cos 0 -}- 1 sin 0)

Thus in Fig. 1 15, p. 285,

the vector AB = 3ar0 = 3(cos 57° + a? sin 57°) = 3(0-545 + 0-8390 = 1-64. -f- 2-5 2*
CD = 41200 = 4(cos 120° + i sin 120°) = — 2 + 3-464*'
EF=2-5sls0 = 2-5cos3i5°-f/ . 2-5 sin 315° = 177 - 1-77*